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Europhysics Letters (EPL)

Discussion on the geometric phase for an open two-level system in the cosmic string spacetime with torsion

Wang Zhi, Chang Xu

Published Jul 10, 2019• 5 citations
Publisher

Abstract

We investigate the geometric phase for a two-level atom interacting with massless scalar field in the background spacetime of a cosmic string with torsion in the framework of open quantum systems. We show that the geometric phase depends on not only the inherent properties of this atom but also on the topological properties of the background spacetime. The correction to the geometric phase of the atom derives from a composite effect of the cosmic string and screw dislocation associated with the curvature and torsion, respectively. The numerical results of the correction to the geometric phase are also present. It is seen that the modification of the phase oscillates around zero and the amplitude of the correction decreases gradually, with the increase of the distance r between the atom and line defects. When the atom gets close to the line defects, the correction will increase significantly but finally tend to a finite value. The behavior of the geometric phase with the variation of the atomic position in three different types of topological defect spacetimes, i.e. , the cosmic string spacetime, screw dislocation spacetime and cosmic string spacetime with torsion, are also shown. We note that the geometric phase in the case of cosmic string spacetime is different from that of other two spacetimes, which suggests that the presence of torsion can significantly affect the geometric phase of the atom, especially in the region near the topological defects. In addition, the geometric phase for the two-level atom which results from the thermal fluctuation of the thermal bath is studied. It is shown that the geometric phase increases with the growth of the temperature T .

DOI
10.1209/0295-5075/126/50005

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We investigate the geometric phase for a two-level atom interacting with massless scalar field in the background spacetime of a cosmic string with torsion in the framework of open quantum systems. We show that the geometric phase depends on not only the inherent properties of this atom but also on the topological properties of the background spacetime. The correction to the geometric phase of the atom derives from a composite effect of the cosmic string and screw dislocation associated with the curvature and torsion, respectively. The numerical results of the correction to the geometric phase are also present. It is seen that the modification of the phase oscillates around zero and the amplitude of the correction decreases gradually, with the increase of the distance r between the atom and line defects. When the atom gets close to the line defects, the correction will increase significantly but finally tend to a finite value. The behavior of the geometric phase with the variation of the atomic position in three different types of topological defect spacetimes, i.e. , the cosmic string spacetime, screw dislocation spacetime and cosmic string spacetime with torsion, are also shown. We note that the geometric phase in the case of cosmic string spacetime is different from that of other two spacetimes, which suggests that the presence of torsion can significantly affect the geometric phase of the atom, especially in the region near the topological defects. In addition, the geometric phase for the two-level atom which results from the thermal fluctuation of the thermal bath is studied. It is shown that the geometric phase increases with the growth of the temperature T .
Abstract We study the Hawking effect in terms of the geometric phase for a two-level atom coupled to vacuum fluctuations in the background of Kerr black hole in a gedanken experiment. Since the atom interacts with a bath of fluctuating massless scalar field, its geometric phase for the nonunitary evolution turns out to be affected by the spacetime curvature that backscatters the vacuum field modes. We examine the geometric phase for two kinds of vacua, i.e. the Unruh vacuum and the Candelas–Chrzanowski–Howard vacuum. We find that for both of these two cases, the geometric phase of the static atom locating outside the infinite redshift surface (or the stationary atom locating inside the ergosphere) exposes that the atom behaves as if it were immersed in a thermal radiation at the Hawking temperature from the Kerr black hole. Therefore, our results show that one in principle can reveal the Hawking effect in the Kerr black hole by analyzing the correction of the geometric phase as opposed to that in a flat spacetime.
In open quantum systems, we study the geometric phases acquired for a two-level atom coupled to a bath of fluctuating vacuum massless scalar fields due to linear acceleration and circular motion without and with a boundary. In free space, as we amplify acceleration, the geometric phase acquired purely due to linear acceleration case firstly is smaller than the circular acceleration case in the ultrarelativistic limit for the initial atomic state $\ensuremath{\theta}\ensuremath{\in}(0,\frac{\ensuremath{\pi}}{2})\ensuremath{\cup}(\frac{\ensuremath{\pi}}{2},\ensuremath{\pi})$, then equals to the circular acceleration case in a certain acceleration, and finally, is larger than the circular acceleration case. The spontaneous transition rates show a similar feature. This result is different from the case of a bath of fluctuating vacuum electromagnetic fields that has been studied. Considering the initial atomic state $\ensuremath{\theta}\ensuremath{\in}(0,\ensuremath{\pi})$, we find that the geometric phase acquired purely due to linear acceleration always equals to the circular acceleration case for the certain acceleration. The feature implies that, in a certain condition, one can simulate the case of the uniformly accelerated two-level atom by studying the properties of the two-level atom in circular motion. Adding a reflecting boundary, we observe that a larger value of a geometric phase can be obtained compared to the absence of a boundary. Besides, the geometric phase fluctuates along zzz, and the maximum of geometric phase is closer to the boundary for a larger acceleration. We also find that geometric phases can be acquired purely due to the linear acceleration case and circular acceleration case with $\ensuremath{\theta}\ensuremath{\in}(0,\ensuremath{\pi})$ for a smaller zzz.
We have investigated the geometric phase acquired by a uniformly accelerated Unruh-DeWitt detector coupled to vacuum fluctuations of a massless conformal scalar field in anti-de Sitter (AdS) spacetime. Using the open-quantum-system formalism, we calculate the phase under three boundary conditions (Dirichlet, transparent, and Neumann) imposed on the field at the AdS boundary. Our findings reveal a sharp distinction between subcritical and supercritical accelerations. For subcritical accelerations, the detector evolves effectively as an isolated system, and the geometric phase is independent of both the AdS radius and the acceleration. For supercritical accelerations, however, topology-acceleration-induced phase corrections emerge and display pronounced boundary-condition dependence. When the AdS radius is smaller than the detector's proper wavelength, the magnitude of the correction at large accelerations follows the ordering Neumann>>>transparent>>>Dirichlet. Moreover, over a finite interval of the detector's weight parameter, both Dirichlet and Neumann boundary conditions produce a richer peak structure in the phase correction than the transparent case, with the detailed pattern governed by the competition between the acceleration and the detector's energy gap. Finally, for transparent boundary conditions in the supercritical regime, the AdS phase correction closely resembles its de Sitter counterpart.
We develop a general framework for investigating the influence of topological defects on the local characteristics of a quantum scalar field in a warped geometry background. The Ricci tensor and curvature scalar are decomposed into contributions from the warp factor, the radial geometry and the angular defect structure. For an arbitrary curvature coupling parameter, the field equation is separated into independent radial, angular and warp-coordinate parts. A complete set of normalized mode functions is obtained for general values of the angular deficit parameters. The general formalism is applied to several specific cases, such as conformally flat warped spacetimes, generalized cosmic strings, global monopoles and anti-de Sitter (AdS)-type warped geometries. The Hadamard two-point function is then evaluated for a global monopole in AdS spacetime using the obtained mode functions.
We study how a uniform torsion background, modeling a continuous density of screw dislocations and induces effective spatial noncommutativity and reshapes the energy spectrum of a free quantum particle. Within the geometric theory of defects, the metric yields a first-order (magnetic-like) coupling in the transverse dynamics, equivalent to an effective magnetic field BeffB_{eff}Beff​ proportional to pzOmegap_z Omegapz​Omega, where OmegaOmegaOmega encodes the torsion strength. In the strong-coupling (Landau) regime, the planar coordinates obey [x,y] != 0 and the spectrum organizes into Landau-like levels with a slight electric-field-driven tilt and a uniform shift. Thus, increasing OmegaOmegaOmega drives the system continuously toward the familiar Landau problem in flat space, with torsion setting the noncommutativity scale and controlling the approach to the Landau limit.
We study, in the framework of open quantum systems, the time evolution of a circularly accelerated two-level atom coupled in the multipolar scheme to a bath of fluctuating vacuum electromagnetic fields. We find that both the spontaneous transition rates and the geometric phase for a circularly accelerated atom do not exhibit a clear sign of thermal radiation characterized by the Planckian factor in contrast to the linear acceleration case. The spontaneous transition rates and effective temperature of the atom are examined in detail in the ultrarelativistic limit and are shown to be always larger than those in the linear acceleration case with the same proper acceleration. Unlike the effective temperature, the geometric phase is dependent on the initial atomic states. We show that when the polar angle in Bloch sphere, $\ensuremath{\theta}$, that characterizes the initial state of the atom equals $\ensuremath{\pi}/2$, the geometric phases acquired due to circular and linear acceleration are the same. However, for a generic state with an arbitrary $\ensuremath{\theta}$, the phase will be in general different, and then we demonstrate in the ultrarelativistic limit that the geometric phase acquired for the atom in circular motion is always larger than that in linear acceleration with same proper acceleration for $\ensuremath{\theta}\ensuremath{\in}(0,\frac{\ensuremath{\pi}}{2})\ensuremath{\cup}(\frac{\ensuremath{\pi}}{2},\ensuremath{\pi})$.